DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Examples, Arclength Parametrization 3(e) Now consider the twisted cubic in R 3 , illustrated in Figure 1.3, given byα(t) =(t, t 2 ,t 3 ), t ∈ R.Its projections in the xy-, xz-, and yz-coordinate planes are, respectively, y = x 2 , z = x 3 ,and z 2 = y 3 (the cuspidal cubic).(f) Our next example is a classic called the cycloid: Itisthe trajectory of a dot on a rollingwheel (circle). Consider the illustration in Figure 1.4. Assuming the wheel rolls withoutOPtaFigure 1.4slipping, the distance it travels along the ground is equal to the length of the circular arcsubtended by the angle through which it has turned. That is, if the radius of the circle is aand it has turned through angle t, then the point of contact with the x-axis, Q, isat unitsto the right. The vector from the origin to the point P can be expressed as the sum of thePOQCPaCt a cos ta sin tFigure 1.5three vectors −→ OQ, −→ QC, and −→ CP (see Figure 1.5):−→OP = −→ OQ + −→ QC + −→ CPand hence the function=(at, 0)+(0,a)+(−a sin t, −a cos t),α(t) =(at − a sin t, a − a cos t) =a(t − sin t, 1 − cos t),t ∈ Rgives a parametrization of the cycloid.(g) A (circular) helix is the screw-like path of a bug as it walks uphill on a right circular cylinderat a constant slope or pitch. If the cylinder has radius a and the slope is b/a, wecan imaginedrawing a line of that slope on a piece of paper 2πa units long, and then rolling the paperup into a cylinder. The line gives one revolution of the helix, as we can see in Figure 1.6. Ifwe take the axis of the cylinder to be vertical, the projection of the helix in the horizontalplane is a circle of radius a, and so we obtain the parametrization α(t) =(a cos t, a sin t, bt).